发表机构
Korea Institute for Advanced Study（韩国高等研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出参数感知的随机特征哈密顿神经网络，在仅含规则动力学的少量参数数据训练下，成功定性外推到训练未覆盖的广阔混沌海，为哈密顿混沌的涌现预测提供了新方法。
AI 中文摘要
哈密顿动力学的机器学习研究推动了对哈密顿神经网络（HNNs）的兴趣，这类网络将哈密顿运动方程编码到学习架构中。尽管取得了这些进展，但仍不清楚这类网络能否预测训练数据中不存在的动力学 regime，尤其是在观测参数区间之外出现的广阔混沌海。我们使用参数感知的随机特征哈密顿神经网络（RF-HNN）来解决这个问题。RF-HNN仅在不变环面占主导的少量控制参数值数据上训练，就能在未见过的参数值下预测自主长时间动力学，此时相空间混合发展且混沌区域扩大，训练或模型选择未使用该 regime 的数据。该方法在四个二自由度哈密顿族（包括Hénon-Heiles系统）上得到验证。通过庞加莱截面几何和有限时间李雅普诺夫指数，我们表明RF-HNN能重现规则结构的破裂以及混沌区域的涌现和增长，而具有相同哈密顿结构的常规训练HNNs仍过于规则。这些结果表明，决定参数外推的不只是哈密顿结构，还包括拟合的哈密顿在控制参数中的延续方式。据我们所知，这是首次证明学习到的哈密顿能从主要是规则的动力学定性外推到训练中不存在的广阔混沌海。
英文摘要
Machine learning of Hamiltonian dynamics has driven growing interest in Hamiltonian neural networks (HNNs), which encode Hamilton's equations of motion into the learning architecture. Despite this progress, it remains unknown whether such networks can predict dynamical regimes absent from their training data, in particular the broad chaotic sea that emerges beyond the observed parameter interval. We address this question using a parameter-aware random-feature Hamiltonian neural network (RF-HNN). Trained using data from only a small number of control-parameter values at which invariant tori dominate, the RF-HNN predicts autonomous long-time dynamics at unseen parameter values where mixed phase space develops and chaotic regions expand, with no data from that regime used in training or model selection. The method is demonstrated across four two-degree-of-freedom Hamiltonian families, including the Hénon-Heiles system. Using Poincaré-section geometry and finite-time Lyapunov exponents, we show that the RF-HNN reproduces the breakup of regular structures and the emergence and growth of chaotic regions, whereas conventionally trained HNNs with the same Hamiltonian structure remain too regular. These results show that what decides parameter extrapolation is not Hamiltonian structure alone but how the fitted Hamiltonian continues in the control parameter. To our knowledge, this is the first demonstration that a learned Hamiltonian can qualitatively extrapolate from predominantly regular dynamics into a broad chaotic sea absent from training.
